Put a coin on a table and surround it with identical coins, each touching it. Exactly six fit, with nothing to spare. Not approximately six: the centres land on a perfect hexagon, six equilateral triangles meet at the middle, and the fit is exact.
That number is the kissing number in two dimensions. The general question is how many non-overlapping unit spheres can simultaneously touch one more of the same size, and the answer is known exactly in remarkably few dimensions.
What is actually known
In three dimensions the answer is 12, but it was not obvious. Isaac Newton and David Gregory disagreed about it in 1694, Newton saying 12 and Gregory arguing 13 might fit. The problem is that 12 spheres around one leave noticeable slack, enough to make 13 look plausible, and it was not proved until 1953 that the slack can never be gathered into a thirteenth position.
Beyond that the exact answers run out quickly. Dimension 4 is 24, settled in 2003. Dimension 8 is 240 and dimension 24 is 196,560, both of which fall out of exceptionally symmetric lattices and were settled well before dimension 4 was. Everything else, including every dimension between 4 and 8, is known only as a range: a construction giving a lower bound, and a separate argument giving an upper bound, with a gap in between.
Eleven dimensions
Dimension 11 sat in that uncomfortable middle, with the best known arrangement containing 592 spheres. In May 2025, as part of the AlphaEvolve results, a configuration of 593 was found. One more sphere.
It is worth being precise about what that is and is not. It is a lower bound: somebody exhibited an arrangement, and the arrangement can be checked. It does not determine the kissing number in eleven dimensions, which remains unknown, and it does not close the gap to the upper bound. Constructing arrangements is the tractable half of the problem. Proving that no better arrangement exists is the hard half, and that is where dimension 3 took 259 years.
This is also why a search procedure is well suited to it. Finding a configuration is a high-dimensional optimisation problem with a checkable answer, which is exactly the shape of problem an evolutionary search handles well. Proving an upper bound is not that shape at all.
The case you can actually compute
The two-dimensional version is not just the easy case, it is the one with a clean formula. Two touching circles have centres R + r apart, and a surrounding circle of radius r blocks off an angle of 2 arcsin(r / (R + r)) at the centre. Divide 360 degrees by that and round down.
Set the radii equal and the angle comes to exactly 60 degrees, six of which is exactly 360, which is the hexagon. Make the surrounding circles smaller and the tidiness disappears: nine circles of radius 5 fit around a circle of radius 10, using 38.94 degrees each and leaving 9.52 degrees that cannot be used for anything. The circles around a circle calculator does this for any pair of radii, and reports the slack, which is the figure that tells you whether a slightly smaller circle would let another one in.
That leftover angle is the intuition for why high dimensions are hard. Past the plane, the arrangement never fits tidily, the slack is distributed in ways that are difficult to account for, and ruling out one extra sphere means ruling out every way of redistributing it.
For the matrix result from the same system, see AlphaEvolve and Strassen.